A shift in the Strauss exponent for semilinear wave equations with a not effective damping

• We consider semilinear wave equations with a not effective damping term.
• We prove a shift in the Strauss exponent for small data global existence.
• We use Klainerman vector fields to prove the existence result in space dimension 2.
• We use radial data and a technique introduced by F. Asakura in space dimension 3.
• We extend to our model the classical blow-up technique due to R.T. Glassey.

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namelyvtt−△v+21+tvt=|v|p,v(0,x)=v0(x),vt(0,x)=v1(x), where p>1p>1, n≥2n≥2. We prove blow-up in finite time in the subcritical range p∈(1,p2(n)]p∈(1,p2(n)] and existence theorems for p>p2(n)p>p2(n), n=2,3n=2,3. In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture p2(n)=p0(n+2)p2(n)=p0(n+2) for n≥2n≥2, where p0(n)p0(n) is the Strauss exponent for the classical semilinear wave equation with power nonlinearity.